On chirality of toroidal embeddings of polyhedral graphs

Abstract

ABSTRACT. We investigate properties of spatial graphs on the standard torus. It is known that nontrivial embeddings of planar graphs in the torus contain a nontrivial knot or a nonsplit link due to [1], [2]. Building on this and using the chirality of torus knots and links [3],[4], we prove that nontrivial embeddings of simple 3-connected planar graphs in the standard torus are chiral. For the case that the spatial graph contains a nontrivial knot, the statement was shown by Castle et al [5]. We give an alternative proof using minors instead of the Euler characteristic. To prove the case in which the graph embedding contains a nonsplit link, we show the chirality of Hopf ladders with at least three rungs, thus generalising a theorem of Simon [6]. topological graphs; knots and links; chirality; topology and chemistry; templating on a toroidal substrate

There exist no minimally knotted planar spatial graphs on the torus

“…Explicit ambient isotopies that transform spatial graphs that fulfil the assumptions of Theorem 1 into the plane R 2 , are given in [13]. Another consequence of Theorem 1 that is stated in the remark has been shown in [11] together with [14]: Nontrivial 3-connected and simple planar spatial graphs that are embedded on a torus are chiral. A graph is simple if it contains no loops and no multi-edges.…”

“…It has been shown in [11] together with [14] that every nontrivial embedding of a simple 3-connected spatial graph on the torus that contains a nontrivial knot or a nonsplit link is chiral. The following remark is therefore a consequence of Theorem 1.…”

Symmetric tangled Platonic polyhedra

“…Tangled embeddings of polyhedral nets { f , z } are less symmetric than their untangled, regular analogs. Earlier studies of tangled nets of the tetrahedron, octahedron, and cube, generated as reticulations of the relevant { f , z } nets on the torus, established that all such “toroidal polyhedra” are topologically chiral ( 26 , 27 ), allowing a pair of distinct isotopes related to each other by a reflection. Further, those toroidal polyhedra are, without exception, rather asymmetric compared with their untangled embeddings.…”